Integrand size = 21, antiderivative size = 18 \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{5/2}}{5 a c} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5004} \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{5/2}}{5 a c} \]
[In]
[Out]
Rule 5004
Rubi steps \begin{align*} \text {integral}& = \frac {2 \arctan (a x)^{5/2}}{5 a c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{5/2}}{5 a c} \]
[In]
[Out]
Time = 4.58 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2 \, \arctan \left (a x\right )^{\frac {5}{2}}}{5 \, a c} \]
[In]
[Out]
\[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
[In]
[Out]
Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2 \, \arctan \left (a x\right )^{\frac {5}{2}}}{5 \, a c} \]
[In]
[Out]
Time = 0.46 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{5\,a\,c} \]
[In]
[Out]